Suppose to have an optimal transportation problem on the product of two Polish spaces $(X,d_X),(Y,d_Y)$ with cost on $Z:=X\times Y$ that is $c=\big(d_X^2-d_Y^2\big)^p$, where $p\in(0,1)$, i.e. you want to find the solution to the problem$$\inf_{\boldsymbol{\pi}\in\Pi({\mu,\nu})}\int_{Z^2}c(z,z')\,\mathrm{d}\boldsymbol{\pi}(z,z')$$where i wrote $z=(x,y)$.
If the $\mu,\nu$ are product measures on the product space $Z$ and the cost is $\boldsymbol{c=d_Z^2=d_X^2+d_Y^2}$, say $\mu=\mu^X\otimes\mu^Y$ and $\nu=\nu^X\otimes\nu^Y$ then you can build an optimal transport plan $\boldsymbol{\pi}$ via a product between the optimal plans $\boldsymbol{\pi}^X$ between $\mu^X,\nu^X$ and $\boldsymbol{\pi}^Y$ between $\mu^Y,\nu^Y$.
In my case with $c=\big(d_X^2-d_Y^2\big)^p, p\in(0,1)$, the optimal plan exists nevertheless but I want a way to build it as a product measure (for $\mu,\nu$ still product measures). Do you have any hints on how I could build an optimal plan using the plans $\boldsymbol{\pi}^X$ and $\boldsymbol{\pi}^Y$? There are some sources about optimal transport on product spaces?
P.s.: every measure here is a probability measure!